Given an embedding of manifolds , the Thom collapse map is a useful approximation to its would-be left inverse. It is used to define pushforward of cohomology-classes along (“Umkehr maps”). It also appears as the key step in Thom's theorem.
Consider and two manifolds and
for the normal bundle;
for the Thom space of the normal bundle;
for any choice of tubular neighbourhood of .
The collapse map (or the Pontrjagin-Thom construction) associated to and the choice of tubular neighbourhood is
Since in the construction of remark 1 every point of is associated to a particular point of , the collapse map lifts to a map
(e.g. Rudyak 98, p. 317)
Of particular interest is the case where in the above is a Cartesian space or rather its one-point compactification, the sphere . By the Whitney embedding theorem, for every there exists an such that every manifold of dimension has an embedding . In this case the collapse map of def. 1 has the form
Composing this further with the canonical map to the universal vector bundle of rank yields a map
from to the th space in the Thom spectrum . This hence defines an element in the homotopy group of the Thom spectrum. Thom's theorem says that all elements in the homotopy groups of arise this way, and that they retain precisely the information of the cobordism equivalence class of manifolds .
In this case the refined Thom collapse map of def. 2 is of the form
Equivalently, one may proceed as follows. For a framed manifold i.e. a manifold with a chosen trivialization of the normal bundle in some one has where is the union of with a disjoint base point. Identify a sphere with a one-point compactification . Then the Pontrjagin-Thom construction is the map obtained by collapsing the complement of the interior of the unit disc bundle to the point corresponding to and by mapping each point of to itself. Thus to a framed manifold one associates the composition
and its homotopy class defines an element in .
with the abstract dual morphisms
The image of this under the -cohomology functor produces
that pushes the -cohomology of to the -cohomology of the point. Analogously if instead of the terminal map we start with a more general map .
More generally a Thom isomorphism may not exists, but may still be equivalent to a twisted cohomology-variant of , namely to , where is an (flat) -(∞,1)-module bundle on and and is the (∞,1)-colimit (the generalized Thom spectrum construction). In this case the above yields a twisted Umkehr map.
The following terms all refer to essentially the same concept:
Stanley Kochmann, section 1.5 of Bordism, Stable Homotopy and Adams Spectral Sequences, AMS 1996
An illustration is given on slide 15
Victor Snaith, Stable homotopy around the arf-Kervaire invariant, Birkhauser 2009
The general abstract formulation in stable homotopy theory is in sketched in section 9 of
and is in section 10 of
with an emphases on parameterized spectra.